Rad HAZU, Matematičke znanosti, Vol. 18 (2014), 73-89.

GENERALIZATIONS AND IMPROVEMENTS OF AN INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA

Sadia Khalid and Josip Pečarić

Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan
e-mail: saadiakhalid176@gmail.com

Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a 10000 Zagreb, Croatia
e-mail: pecaric@element.hr

Abstract.   Some generalizations of an inequality of Hardy-Littlewood-Pólya are presented. We discuss the n-exponential convexity and log-convexity of the functions associated with the linear functional defined by the generalized inequality and also prove the monotonicity property of the generalized Cauchy means obtained via this functional. Finally, we give several examples of the families of functions for which the results can be applied.

2010 Mathematics Subject Classification.   26A48, 26A51, 26D15.

Key words and phrases.   Non-increasing sequence in mean, Wright-convex function, n-exponential and logarithmic convexity, divided difference.

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